Begovic tweeted he's delighted about moving to the Serie A giants:
The 32-year-old was part of Chelsea's 2016-17 Premier League winning squad, operating as Thibaut Courtois' deputy.
Begovic left Stamford Bridge in 2017 to become the Cherries' No. 1, but the keeper fell out of favour with manager Eddie Howe.
The player was on loan with Azerbaijan Premier League champions Qarabag, but had recently returned to his parent club in England.
Per Sky Sports, Howe said he was happy with how Begovic was training after he arrived back in Bournemouth, but admitted his stay with the squad might be short.
The Bosnia and Herzegovina international will now remain with Milan until the summer as Gianluigi Donnarumma's back-up, and could be in line for a quick debut.
Football Italia reported Donnarumma is a doubt for Wednesday's Coppa Italia match against Serie A bottom side SPAL because of a thigh injury.